mathematics

Sangaku

Sangaku is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sangaku rather than just read about it. In short: Sangaku or san gaku (Japanese: 算額, lit. 'calculation tablet') are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes. History The sangaku were painted in color on wooden tablets (ema) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas…

Key takeaways

Sangaku belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sangaku to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sangaku from memory before moving on to harder problems.

Reference excerpt

Sangaku or san gaku (Japanese: 算額, lit. 'calculation tablet') are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.

History

The sangaku were painted in color on wooden tablets (ema) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas, as challenges to the congregants, or as displays of the solutions to questions. Many of these tablets were lost during the period of modernization that followed the Edo period, but around nine hundred are known to remain. Fujita Kagen (1765–1821), a Japanese mathematician of prominence, published the first collection of sangaku problems, his Shimpeki Sampo (Mathematical problems Suspended from the Temple) in 1790, and in 1806 a sequel, the Zoku Shimpeki Sampo. During this period Japan applied strict regulations to commerce and foreign relations for western countries so the tablets were created using Japanese mathematics, developed in parallel to western mathematics. For example, the connection between an integral and its derivative (the fundamental theorem of calculus) was unknown, so sangaku problems on areas and volumes were solved by expansions in infinite series and term-by-term calculation.

Select examples

A typical problem, which is presented on an 1824 tablet in Gunma Prefecture, covers the relationship of three touching circles with a common tangent, a special case of Descartes' theorem. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is:

1 r middle = 1 r left + 1 r right . {\displaystyle {\frac {1}{\sqrt {r_{\text{middle}}}}}={\frac {1}{\sqrt {r_{\text{left}}}}}+{\frac {1}{\sqrt {r_{\text{right}}}}}.}

(See also Ford circle.)

Soddy's hexlet, thought previously to have been discovered in the west in 1937, had been discovered on a sangaku dating from 1822. One sangaku problem from Sawa Masayoshi and other from Jihei Morikawa were solved only recently.

See also Equal incircles theorem Japanese theorem for concyclic polygons Japanese theorem for concyclic quadrilaterals Problem of Apollonius Recreational mathematics Seki Takakazu

Notes

References Fukagawa, Hidetoshi, and Dan Pedoe. (1989). Japanese temple geometry problems = Sangaku. Winnipeg: Charles Babbage. ISBN 9780919611214; OCLC 474564475 __________ and Dan Pedoe. (1991) How to resolve Japanese temple geometry problems? (日本の幾何ー何題解けますか?, Nihon no kika nan dai tokemasu ka) Tōkyō : Mori Kitashuppan. ISBN 9784627015302; OCLC 47500620 __________ and Tony Rothman. (2008). Sacred Mathematics: Japanese Temple Geometry. Princeton: Princeton University Press. ISBN 069112745X; OCLC 181142099 Huvent, Géry. (2008). Sangaku. Le mystère des énigmes géométriques japonaises. Paris: Dunod. ISBN 9782100520305; OCLC 470626755 Rehmeyer, Julie, "Sacred Geometry", Science News, March 21, 2008. Rothman, Tony; Fugakawa, Hidetoshi (May 1998). "Japanese Temple Geometry". Scientific American. pp. 84–91.

External links

Sangaku (Japanese votive tablets featuring mathematical puzzles) Japanese Temple Geometry Problem Sangaku: Reflections on the Phenomenon Sangaku Journal of Mathematics

Illustrations

Worked examples

Example 1 — a first encounter with Sangaku

Start with the simplest possible case. Write down what Sangaku claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sangaku before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sangaku ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sangaku

In research: Sangaku appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sangaku in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sangaku is common in secondary-school and first-year university syllabi. It links to neighbouring topics Euclidean geometry, Japanese mathematics, Recreational mathematics, so understanding it makes those chapters shorter.
In everyday life: Look for Sangaku outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sangaku in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sangaku means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sangaku out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sangaku in simple terms?

Why does Sangaku matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sangaku?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sangaku.